Randomized Oversampling for Generalized Multiscale Finite Element Methods

Handle URI:
http://hdl.handle.net/10754/608586
Title:
Randomized Oversampling for Generalized Multiscale Finite Element Methods
Authors:
Calo, Victor M. ( 0000-0002-1805-4045 ) ; Efendiev, Yalchin R. ( 0000-0001-9626-303X ) ; Galvis, Juan; Li, Guanglian
Abstract:
In this paper, we develop efficient multiscale methods for flows in heterogeneous media. We use the generalized multiscale finite element (GMsFEM) framework. GMsFEM approximates the solution space locally using a few multiscale basis functions. This approximation selects an appropriate snapshot space and a local spectral decomposition, e.g., the use of oversampled regions, in order to achieve an efficient model reduction. However, the successful construction of snapshot spaces may be costly if too many local problems need to be solved in order to obtain these spaces. We use a moderate quantity of local solutions (or snapshot vectors) with random boundary conditions on oversampled regions with zero forcing to deliver an efficient methodology. Motivated by the randomized algorithm presented in [P. G. Martinsson, V. Rokhlin, and M. Tygert, A Randomized Algorithm for the approximation of Matrices, YALEU/DCS/TR-1361, Yale University, 2006], we consider a snapshot space which consists of harmonic extensions of random boundary conditions defined in a domain larger than the target region. Furthermore, we perform an eigenvalue decomposition in this small space. We study the application of randomized sampling for GMsFEM in conjunction with adaptivity, where local multiscale spaces are adaptively enriched. Convergence analysis is provided. We present representative numerical results to validate the method proposed.
KAUST Department:
Numerical Porous Media SRI Center (NumPor)
Citation:
Randomized Oversampling for Generalized Multiscale Finite Element Methods 2016, 14 (1):482 Multiscale Modeling & Simulation
Publisher:
Society for Industrial & Applied Mathematics (SIAM)
Journal:
Multiscale Modeling & Simulation
Issue Date:
23-Mar-2016
DOI:
10.1137/140988826
Type:
Article
ISSN:
1540-3459; 1540-3467
Sponsors:
Yalchin Efendiev would like to thank the partial support from the U.S. Department of Energy Office of Science, Office of Advanced Scientific Computing Research, Applied Mathematics program under Award DE-FG02- 13ER26165 and the DoD Army ARO Project.
Additional Links:
http://epubs.siam.org/doi/10.1137/140988826
Appears in Collections:
Articles

Full metadata record

DC FieldValue Language
dc.contributor.authorCalo, Victor M.en
dc.contributor.authorEfendiev, Yalchin R.en
dc.contributor.authorGalvis, Juanen
dc.contributor.authorLi, Guanglianen
dc.date.accessioned2016-05-08T14:16:58Zen
dc.date.available2016-05-08T14:16:58Zen
dc.date.issued2016-03-23en
dc.identifier.citationRandomized Oversampling for Generalized Multiscale Finite Element Methods 2016, 14 (1):482 Multiscale Modeling & Simulationen
dc.identifier.issn1540-3459en
dc.identifier.issn1540-3467en
dc.identifier.doi10.1137/140988826en
dc.identifier.urihttp://hdl.handle.net/10754/608586en
dc.description.abstractIn this paper, we develop efficient multiscale methods for flows in heterogeneous media. We use the generalized multiscale finite element (GMsFEM) framework. GMsFEM approximates the solution space locally using a few multiscale basis functions. This approximation selects an appropriate snapshot space and a local spectral decomposition, e.g., the use of oversampled regions, in order to achieve an efficient model reduction. However, the successful construction of snapshot spaces may be costly if too many local problems need to be solved in order to obtain these spaces. We use a moderate quantity of local solutions (or snapshot vectors) with random boundary conditions on oversampled regions with zero forcing to deliver an efficient methodology. Motivated by the randomized algorithm presented in [P. G. Martinsson, V. Rokhlin, and M. Tygert, A Randomized Algorithm for the approximation of Matrices, YALEU/DCS/TR-1361, Yale University, 2006], we consider a snapshot space which consists of harmonic extensions of random boundary conditions defined in a domain larger than the target region. Furthermore, we perform an eigenvalue decomposition in this small space. We study the application of randomized sampling for GMsFEM in conjunction with adaptivity, where local multiscale spaces are adaptively enriched. Convergence analysis is provided. We present representative numerical results to validate the method proposed.en
dc.description.sponsorshipYalchin Efendiev would like to thank the partial support from the U.S. Department of Energy Office of Science, Office of Advanced Scientific Computing Research, Applied Mathematics program under Award DE-FG02- 13ER26165 and the DoD Army ARO Project.en
dc.language.isoenen
dc.publisherSociety for Industrial & Applied Mathematics (SIAM)en
dc.relation.urlhttp://epubs.siam.org/doi/10.1137/140988826en
dc.rightsArchived with thanks to Multiscale Modeling & Simulationen
dc.subjectgeneralized multiscale finite element methoden
dc.subjectoversamplingen
dc.subjecthigh contrasten
dc.subjectrandomized approximationen
dc.subjectsnapshot spaces constructionen
dc.titleRandomized Oversampling for Generalized Multiscale Finite Element Methodsen
dc.typeArticleen
dc.contributor.departmentNumerical Porous Media SRI Center (NumPor)en
dc.identifier.journalMultiscale Modeling & Simulationen
dc.eprint.versionPublisher's Version/PDFen
dc.contributor.institutionApplied Geology Department, Western Australian School of Mines, Faculty of Science and Engineering, Curtin University, Perth, WA, Australiaen
dc.contributor.institutionDepartment of Mathematics & Institute for Scientific Computation (ISC), Texas A&M University, College Station, TXen
dc.contributor.institutionDepartamento de Matematicas, Universidad Nacional de Colombia, Bogota D.C., Colombiaen
dc.contributor.institutionInstitute for Numerical Simulation, University of Bonn, 53115 Bonn, Germanyen
dc.contributor.affiliationKing Abdullah University of Science and Technology (KAUST)en
kaust.authorCalo, Victor M.en
kaust.authorEfendiev, Yalchin R.en
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